Kavli Affiliate: Pau Amaro Seoane
| First 5 Authors: Pau Amaro Seoane, Pau Amaro Seoane, , ,

| Summary:
We demonstrate the necessity of fractional calculus for modeling anomalous diffusion in stellar dynamics, specifically Resonant Relaxation (RR) near a supermassive black hole (SMBH). Standard transport theories rely on the local Fokker-Planck (FP) equation, a formalism restricted to Gaussian processes governed by the Central Limit Theorem (CLT). This approach fails for RR, which is a superdiffusive Lévy flight characterized by non-local transport and governed by the Generalized Central Limit Theorem (GCLT). We apply the space-fractional Fokker-Planck equation (FFPE), utilizing non-local Riemann-Liouville operators, to resolve major discrepancies between local theory and observations. In transient regimes, the FFPE predicts an immediate, linear flux ($Γ(t) propto t$), consistent with high Tidal Disruption Event (TDE) rates in post-starburst galaxies, whereas local FP models predict an unphysical exponential delay ($Γ(t) propto exp(-C/t)$). Furthermore, we prove mathematically that the non-local integral operators permit “barrier jumping”. This mechanism nullifies transport bottlenecks like the proposed Schwarzschild Barrier (SB). Local models incorrectly interpret the SB as severely suppressing Extreme Mass-Ratio Inspiral (EMRI) rates. We validate our theoretical findings using direct-summation $N$-body simulations incorporating relativistic corrections. The simulations confirm that particle flux remains continuous across the SB in the regime where RR dominates, demonstrating that the barrier does not inhibit EMRI production. The fractional framework provides the correct physical description for non-local transport, resolving long-standing problems concerning TDE and EMRI rates.
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